Will Bergman and Mike Ma Harmonic Oscillators Will Bergman and Mike Ma

Overview Simple Harmonic Motion Driven Simple Harmonic Oscillators Spring and Mass Periodic Driving Force General Form Resonance Damped Simple Harmonic Oscillators Tacoma Bridge Example Conclusion and Further Applications Underdamped Case Overdamped Case Critically Damped Case

Spring and Mass 𝐹=𝑚𝑎 𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥 Guess: 𝑥 𝑡 = 𝑒 𝑟𝑡 𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥 Guess: 𝑥 𝑡 = 𝑒 𝑟𝑡 𝑟 1 =+𝑖 𝑘 𝑚 , 𝑟 2 =−𝑖 𝑘 𝑚 𝑥 𝑡 = 𝐶 1 𝑥 1 𝑡 + 𝐶 2 𝑥 2 𝑡 𝑥 𝑡 = 𝐶 1 𝑒 𝑟 1 𝑡 + 𝐶 2 𝑒 𝑟 1 𝑡 (Taylor, 2003) https://commons.wikimedia.org/wiki/File:Simple_harmonic_oscillator.gif

General Form 𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥 𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥 𝑥 𝑡 = 𝐶 1 + 𝐶 2 𝑐𝑜𝑠 𝑤𝑡 +𝑖 𝐶 1 − 𝐶 2 𝑠𝑖𝑛 𝑤𝑡 𝑥 𝑡 =𝐵 1 𝑐𝑜𝑠 𝑤𝑡 + 𝐵 2 𝑠𝑖𝑛 𝑤𝑡 (Taylor, 2003)

Damped Simple Harmonic Oscillators B – damping constant 𝑤 0 - natural frequency Relationship between B and 𝑤 0 determine different cases of damping Solution form: 𝑥 𝑡 = 𝑒 𝑟𝑡

Underdamped Case (𝐵< 𝑤 0 ) 𝐵 2 − 𝑤 0 2 =𝑖 𝑤 0 2 − 𝐵 2 =𝑖 𝑤 1 𝑥 𝑡 =𝑒 −𝐵𝑡 ( 𝐶 1 𝑒 𝑖 𝑤 1 𝑡 + 𝐶 2 𝑒 −𝑖 𝑤 1 𝑡 ) Amplitude of oscillations decrease exponentially (Taylor, 2003)

Overdamped Case (𝐵> 𝑤 0 ) 𝑥 𝑡 = 𝑒 −𝐵𝑡 ( 𝐶 1 𝑒 𝐵 2 − 𝑤 0 2 𝑡 + 𝐶 2 𝑒 − 𝐵 2 − 𝑤 0 2 𝑡 ) No Oscillations! (Taylor, 2003)

Critically Damped Case (𝐵= 𝑤 0 ) Repeated Eigenvalues 𝐵= 𝑤 0 is a bifurcation value 𝑥 𝑡 = 𝑒 −𝐵𝑡 𝑥(𝑡)=𝑡𝑒 −𝐵𝑡 𝑥 𝑡 = 𝐶 1 𝑒 −𝐵𝑡 + 𝐶 2 𝑡𝑒 −𝐵𝑡 (Blanchard et al., 2012) (Taylor, 2003)

Driven Simple Harmonic Oscillators 𝑑 2 𝑥 𝑑 𝑡 2 +𝐵 𝑑𝑥 𝑑𝑡 + 𝑤 0 2 𝑥=𝑓(𝑡) Solution = general solution of homogeneous equation (unforced) + one particular solution to nonhomogeneous equation (forced) 𝑥 𝑡 = 𝐶 1 𝑥 1 𝑡 + 𝐶 2 𝑥 2 𝑡 + 𝑥 𝑝 (𝑡) Resonance- the frequency of the driving force is equal to the natural frequency of the oscillating system http://www.acs.psu.edu/drussell/Demos/SHO/mass-force.html

Conclusion and Further Applications of Theory https://www.youtube.com/watch?v=vPZuHFrawz4 https://www.youtube.com/watch?v=3mclp9QmCGs

References https://commons.wikimedia.org/wiki/File:Simple_harmonic_oscillator.gif Taylor, John R. "Chapter 5: Oscillations." Classical Mechanics. Sausalito, CA: U Science, 2005. 161-203. Print. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. "Chapter 2.3: The Damped Harmonic Oscillator." Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, 2012. 183-88. Print. http://www.acs.psu.edu/drussell/Demos/SHO/mass-force.html